Microfluidics and Dimensionless Numbers

Size is important. As the size of our system changes, so do its physical properties. Welcome to the world of dimensionless numbers and fluids on a micro scale. We were forced into this scale because shooting anything into space requires careful evaluation of necessary mass, every gram in payload costs a ton, but when you start looking into the beauty of what happens on such a small scale, you will get hooked like I have. 

In the word of small scale, surface area to volume ratio becomes smaller, so all of those molecular forces and charge distributions across a molecule you were told to just ignore in chemistry class come to bear. To understand when fluids will respond to bulk forces vs molecular-molecular forces, we have a couple of numbers, dimensionless number that help determine what behaviour we are likely to see.  To frame this discussion, let's briefly discuss fluids and how they are characterized. Fluids are continuum materials. That is they have discreet properties, such as mass and force, that are observed in bulk scale as measureable parameters in relation to volume, such as density and force density. The forces that act on fluids are both fluid stresses due to intermolecular forces with respect to surface area or bodily forces that respond generally to the mass of the molecule or the fluid as a body. The take home message is that microfluidics allows us to manipulate nature on a molecule by molecule scale, since in microfluidics, as I've learned, the properties of the molecules matter a lot. And these properties can be explored and exploited.

There are a lot of numbers to consider, but we will focus on those that will apply to our project. Those numbers are Reynold's (Re), Pe'clet's (Pe), and Capillary Number (Ca). I won't get into the nitty-gritty of these numbers in this post. (I will probably wait until we have begun to play with and get frustrated with the concepts that these numbers define to discuss how cool and inconvenient the properties of fluid flow on the microscale can be). Instead, I will talk about the basic concept each number represents and how we will capitalize and be challenged by each number's meaning. 

Reynold's number (Re) describes the boundary between viscous and inertial flow, ie how much molecules stick together because of intermolecular forces and how much the fluid will flow with the bulk due to velocity of the flow. What we see is, as a consequence of molecular forces overcoming bulk flow, is laminar flow, that is flow without turbidity, where two adjacent fluids simply don't mix. Molecules in the flow carry on in a straight path, while molecules overwhelmed by molecular forces are often slowed by the attraction to the channel walls. For us, any channel smaller than 1mm in width will exhibit laminar flow instead of turbid flow as indicated by the low Reynold's number and whose properties can be predicted by the linear Stokes equation. Turbidity  means mixing. So, if we do want molecules to mix, like our lysing solution, we could implement some tight turns in our channels or rely on diffusion. Other solutions will be explored in later posts.

Pe'clet's number (Pe) describes the boundary between diffusive and convectional forces. And this is where microfluidics begins to get interesting. Convection transfer force at the tangent to fluid flow, In laminar flow, the the tangent to the flow is hard to come by. So, diffusion, the mixing of particles due to a concentration gradient, is the sole mixing force. But, the trade-off is that it becomes very easy to separate molecules to a fine degree by only allowing a certain amount of time for the particles with the greatest diffusion to do so. Hence, we would solve for the channel length necessary to filter out mRNA from the crude cell lysate. So low Pe numbered solutes will diffuse, while the bulk, with much higher Pe numbers, will stay in their home laminar stream (ie will take much more time (as described by distance) to diffuse).

Capillary number (Ca) describes the boundary between viscous and interfacial forces. The interfacial force appears between the wall and the fluid or between two immiscible fluids (which is utilized in droplet microfluidics), The viscous forces, preventing flow, are between the molecules within the fluid. We use this property to move fluids throughout the device without pumps. The fluid can be attracted to move because the surface of the channel is more chemically or electrically compatible than the viscous forces between the fluid molecules. As the fluid molecules creep at the fluid front, the adjacent molecules are pulled along due to viscous forces. Generally motion happens at Ca about 1 or less.

There are many other dimensionless numbers and you can look more into them by checking out this paper : http://faculty.washington.edu/amyshen/ME599G/squiresquake.pdf